To model agents adopting a social norm, we extend the set of behavioural strategies to allow for players to condition their actions on the average reputation of the other players within their group.
We condition actions on the average reputation firstly because a player's payoff is dependent on the actions of his group members and secondly because directly observing the actions of others can be difficult in sizeable groups, whereas observing the overall sentiment is arguably easier. Conditioning actions to the average reputation is also a key difference between our setup and [1]} where actions depend on levels of cooperation, and from [2]}, where they depend on the amounts contributed. Indeed, given that there are three types of actions, each receiving a reputation on the basis of a shared social norm, reputational information can only be used as a signal of the cooperativeness of the environment a player is facing, similar to the case of image scoring. Furthermore, in our model reputation contains both elements of direct and indirect reciprocity as when groups are reshuffled, reputations are carried over. Finally, we significantly extend the strategy set considering strategies whose primary action in a benign environment is not cooperating. The strategy X\(^{\mbox{\tiny $k_{\text{min}}$,Y}}_{\mbox{\tiny $\text{Z}_\text{C}\text{Z}_\text{D}\text{Z}_\text{L}$}}\)  defines an agent taking action X \(\in \lbrace \text{C,D,L}\rbrace \)  if the average reputation of the other members of the group strictly exceeds \(k_{\text{min}} \in [-1, 1)\) , and Y \(\in \lbrace \text{C,D,L}\rbrace \)  otherwise. Furthermore, the subscripts describe the agent's strategy in terms of its punishment decision taken against cooperators ZC, defectors ZD, and loners ZL, where ZC, ZD, ZL \(\in \lbrace \text{P,N}\rbrace \)  is the decision to punish or not to punish respectively. In addition to the pure strategies of being an unconditional cooperator, defector and loner, this adds a further 8 strategies: C\(^{\mbox{\tiny -1,D}}_{\mbox{\tiny }}\) , C\(^{\mbox{\tiny 0,D}}_{\mbox{\tiny }}\) , C\(^{\mbox{\tiny -1,L}}_{\mbox{\tiny }}\) , C\(^{\mbox{\tiny 0,L}}_{\mbox{\tiny }}\) , D\(^{\mbox{\tiny -1,L}}_{\mbox{\tiny }}\) , D\(^{\mbox{\tiny 0,L}}_{\mbox{\tiny }}\) , L\(^{\mbox{\tiny -1,D}}_{\mbox{\tiny }}\) , L\(^{\mbox{\tiny 0,D}}_{\mbox{\tiny }}\) , each of which have 8 variants prescribing punishment towards zero, one or more cooperators, defectors and loners. In an effort to minimise the size of the strategy space, and therefore also some of the effects of random drift, we remove counterintuitive strategies, i.e. those that would cooperate only in groups of defectors or loners, defecting or going loner when surrounded by cooperators. For the same reason, strategies of higher complexity, i.e. those implying multiple behavioural shifts and/or more than two possible actions, are excluded from the present analysis and relegated to further studies.
